The more information available about a user, the easier it is for him to be trapped in a lie.

The parameter \(\rho \)  in Equation REF  is a discount factor chosen in the set \((0,1]\) . It is worth noting that \(\rho \)  is only important if we wish to consider the limiting dynamics as \(T \rightarrow \infty \) . When \(\rho < 1\) , the user recognizes that future rewards have less value than rewards more immediately. To relate the parameters \(x^{(j)}\) , \(y^{(j)}\)  and \(z^{(j)}\)  to Equation REF , we need to compute the expected value \(\mathbb {E}\left(\Pi ^{(j)}\right)\) . In the form given, this may be complex, since the functions defined in Equation REF  maybe non-linear, meaning we cannot pass the expectation operator through the expression.
The solution to the game is then defined by the simultaneous optimization problem:
\(\forall j \left\lbrace \begin{aligned}\max \;\; & \mathbb {E}\left(\Pi ^{(j)}(\mathbf {x}(t),\mathbf {y}(t),\mathbf {z}(t))\right)\\s.t. \;\; & x^{(j)}(t) + y^{(j)}(t) + z^{(j)}(t) = 1 \quad \forall t\\& x^{(j)}(t), y^{(j)}(t), z^{(j)}(t) \ge 0 \quad \forall t\end{aligned}\right.\) 
where \(\mathbf {x}(t)\) , \(\mathbf {y}(t)\) , \(\mathbf {z}(t)\)  are the vectors of decision variables for the players. Let
\(\Omega = \prod _{j,t} \left\lbrace (x^{(j)}(t),y^{(j)}(t),z^{(j)}(t)) \in [0,1]^3 :\right.\\x^{(j)}(t) + y^{(j)}(t) + z^{(j)}(t) = 1, \\\left.x^{(j)}(t), y^{(j)}(t), z^{(j)}(t) \ge 0 \right\rbrace \) 
This is the complete strategy space for all players over the course of time \(t \in [0,T]\) . Any Nash equilibrium will be chosen from this strategy space. Theorem 1 of [1]} provides the following (uninteresting) result:
Proposition 4.1 Suppose that \(\mathbb {E}\left(\Pi ^{(j)}(\mathbf {x}(t),\mathbf {y}(t),\mathbf {z}(t))\right)\)  is concave for all \(j\)  then there is a Nash equilibrium in \(\Omega \)  for this game.

Remark 4.2 The uniqueness of a Nash equilibrium in this case is completely a function of the structure of specific objective functions.

We noted above that the structure of \(\mathbb {E}\left(\Pi ^{(j)}(\mathbf {x}(t),\mathbf {y}(t),\mathbf {z}(t))\right)\)  maybe complex. By way of simplification, we can write a specific form of Equation REF , as:
\(\Pi ^{(j)}(t) = \\w_1\alpha (x^{(j)},z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)},t)\left(X^{(j)}(t)+(1-\tau (t))Z^{(j)}(t)\right) \\+w_2\left(\beta (y^{(j)},\bar{y}^{(j)},t)Y^{(j)}(t) + \eta Z^{(j)}(t)\right) \\-w_3\gamma (z^{(j)},\bar{z}^{(j)},t)\tau (t)Z^{(j)}(t) - w_4\zeta Z^{(j)}(t) \\-w_5\theta X^{(i)}(t)\) 
Here we replace the functions from Equation REF  with piecewise constant multipliers. We can then relate Equation REF  to the parameters \(x^{(j)}\) , \(y^{(j)}\)  and \(z^{(j)}\) , we note that:
\(\mathbb {E}\left(\Pi ^{(j)}\right) =\sum _{t = 0}^{T}\rho ^t \cdot \\ \left[w_1\alpha (x^{(j)},z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)},t)\left(x^{(j)}(t)+(1-\tau (t))z^{(j)}(t)\right) \right.\\+w_2\left(\beta (y^{(j)},\bar{y}^{(j)},t)y^{(j)}(t) + \eta z^{(j)}(t)\right) - \\w_3\gamma (z^{(j)},\bar{z}^{(j)},t)\tau (t)z^{(j)}(t) - \\\left. w_4\zeta z^{(j)}(t) - w_5\theta x^{(i)}(t)\right]\) 
That is, a user's expected payoff after engaging in this game, is a function of the proportion of time he releases information, withholds information and lies about information. Moreover, because we assume the reward/cost multipliers (\(\alpha \) , \(\beta \)  and \(\gamma \) ) are dependent on the group average rates of releasing, withholding and lying about information, the payoff to Player \(j\)  is dependent on the choices of all other players in his circle of friends. Thus, a game dynamic is established. We study this simplified game form in the remainder of the paper.
Evolutionary Dynamics and Example
A critical problem with the game defined in the previos section is that individuals will never optimize their behavior according to it. An individual can estimate many of the parameters in the model, but will never make decisions based on the long run objective of maximizing his utility function. It will simply be impossible for an individual to chose an optimizing strategy ab initio, particularly without having a clear understanding of the strategies of other players. This is especially true if (as is likely the case) multiple Nash equilibria exist.
However, a user may engage in a more evolutionary model of decision making [2]}Weibull's book, [2]} is an introduction to evolutionary game theory, which inspires the approach described herein. We are not proposing a classical evolutionary game. Instead, we are proposing an evolutionary mechanism applied to a game theoretic context that describes user learning.:
At any time \(t\) , Player \(j\)  has a strategy \((x^{(j)}(t),y^{(j)}(t),z^{(j)}(t))\)  with an initial strategy \((x^{(j)}(0),y^{(j)}(0),z^{(j)}(0))\)

At each time \(t\) , Player \(j\)  will solve the one-stage game derived from Equation REF  by finding a maximizing strategy \(\left(\hat{x}^{(j)}, \hat{y}^{(j)}, \hat{z}^{(j)}\right)\)  with respect to the current observed strategies of the other players and the current parameters in the model.

Each player's strategy is updated by the rule:
\(x^{(j)}(t+1) = x^{(j)}(t) + \epsilon ^{(j)}\left(\hat{x}^{(j)} - x^{(j)}(t)\right)\\y^{(j)}(t+1) = y^{(j)}(t) + \epsilon ^{(j)}\left(\hat{y}^{(j)} - y^{(j)}(t)\right)\\x^{(j)}(t+1) = z^{(j)}(t) + \epsilon ^{(j)}\left(\hat{z}^{(j)} - z^{(j)}(t)\right)\) 

Here \(\epsilon ^{(j)}\)  is a learning rate associated to the player, and is assumed to be small – that is, \(\epsilon ^{(j)} \ll 1\) . The dynamics given in Equations REF  -  are a discrete variation of the Jacobi iteration for finding equilibria in games (see e.g., [4]}). At its core, this is just a form of gradient ascent.
Intuitively, each time a user makes a decision about a piece of information, he considers his knowledge of the other players and computes an optimal move for this time period. However, instead of changing his strategy completely, he modifies his strategy (learns) by a small amount in the direction of optimality. This is consistent with an individual who learns the average behavior of the social network.
Example
By way of example, assume we have a small clique of three friends on a social network (this is the graph governing Equation REF ). Consider the following multiplier definitions for use in Equation REF . These functions are derived from a qualitative analysis of the data collected in the experiment described in the previous sections, and are intended to be simple token counting margin functions.
\(\alpha (x^{(j)},z^{(j)},\bar{x}^{(j)},\bar{z}^{(j)}) = {\left\lbrace \begin{array}{ll}1 & x^{(j)}+z^{(j)} \le \bar{x}^{(j)}+\bar{z}^{(j)}\\0 & \text{otherwise}\end{array}\right.}\) 
In this case, the social value of information is non-zero only if the information provided is in some way lower than the mean information provided by the group. That is to say, you can social capital only if you're not posting more information than the group average. However, if the group average is high, you will accrue social capital the more you post.
\(\beta (y^{(j)},\bar{y}^{(j)}) = {\left\lbrace \begin{array}{ll}1 & y^{(j)}\ge \bar{y}^{(j)}\\0 & \text{otherwise}\end{array}\right.}\) 
Here, the privacy value is non-zero only if the amount of information that is to be admitted is larger than average amount of information being admitted. Similarly, we can define:
\(\gamma (z^{(j)},\bar{z}^{(j)}) = {\left\lbrace \begin{array}{ll}1 & z^{(j)}\ge \bar{z}^{(j)}\\0 & \text{otherwise}\end{array}\right.}\) 
In this case, the cost of a lie is only non-zero if the lie is in some sense more egregious than the average level of dishonesty. Finally, we can set: \(\zeta =1\)  and \(\theta =1\) .
Variations in the players behavior can then be created by modifying \(w_1,\dots ,w_5\)  in Equation REF . Finally, we assume that \(\tau \)  increases linearly in time from \(0.1\)  to \(0.9\) . Recall, \(\tau \)  is the probability that a lie can be detected by the social network. Thus, as time proceeds, it becomes more likely that a falsehood is detected because more information is available about each player In a fully formalized model, we believe that \(\tau \)  will be a function of \(Q\)  (defined in Equation REF ), however for our simple studies, this is sufficient.
We study three specific examples of the dynamics produced by this model to illustrate that even these (simple) dynamics are capable of qualitatively reproducing behavior observed in our study. In the first example, social capital is deemed more important than privacy (\(w_1 = 1\) , \(w_2 = 0.25\)  and \(w_5 = 0.125\) ), but there is a stronger sense of morality (\(w_3 = 0.5\) , \(w_4 = 1\) ). We assume three identical players connected by a complete graph with three vertices. Player evolution is illustrated in Figure  REF
<FIGURE>When we start the game with three players, each playing the strategy \(x^{(j)}=0.7\) , \(y^{(j)} = 0.2\)  and \(z^{(j)} = 0.1\) , we see deception is removed from the system relatively quickly, while information hiding increases (replacing deception in the system). Notice the system converges to a stationary strategy near \(x^{(j)}=2/3\) , \(y^{(j)} = 1/3\)  and \(z^{(j)} = 0\) . In simpler terms, these equilibrium points are consistent with our findings that social capital is much more important than the gain associated with privacy, because we see that there is a preference toward sharing information truthfully. This was observed in Section REF . It is also worth noting, that it can be shown numerically these are limiting Nash equilibria for this example.
By way of comparison, we can construct a game with less morality and even more importance associated to social capital (being obtained by any means necessary) with \(w_1 = 2\) , \(w_2 = 0.25\) , \(w_3 = 0.25\) , \(w_4 = 0.125\)  and \(w_5 = 0.125\) . Note, \(w_4 = 0.125\)  indicates a low moral penalty for lying. The evolution of the players is illustrated in Figure REF .
<FIGURE>The dynamics in this game are different, than those of the first game. There is an initial, substantial, increase in the level of deception (when it is easy to lie) in order to obtain social capital. As it becomes more difficult to lie, the players return to a more truthful scenario that is easier to support.
This example confirms the identified effects of the signaling theory in our dataset: users are less likely to deceive when they are heavily involved in social interactions (Section III-C1 and III-C2).
<FIGURE>In our final example, we consider a highly moral player who puts less emphasis on social capital and substantial emphasis on information privacy. In this case, we have: \(w_1 = 0.5\) , \(w_2 = 5\) , \(w_3 = 2\) , \(w_4 = 100\)  and \(w_5 = 3\) . The results are illustrated in Figure REF .
This objective function models a user who is highly moral deciding whether to release an information type that may be sensitive, such as GPA or dating status and illustrates the ability of the model to capture the various qualitative results observed in the survey. In particular, this result is consistent with the finding reported in Section REF : withholding is used as a form of control when deception is considered unethical.
The proposed approach can also be used to study richer scenarios, including those with players that begin with different strategies and games that are played on distinct graph structures as was discussed in [5]}.

Future Directions
In this work, we have shown an informed model on deception and misrepresentation in OSNs. The model is derived from a token counting approach modeled in a stochastic, continuous Petri net. This net is then used to derive an objective function for each player in which the payoff to a given player is a function not only of his decisions but also the decisions of his circle of friends. This leads to a game theoretic framework. We show that while this game has at least one Nash equilibria, it is more interesting to consider an evolutionary game dynamic in which players learn over time and converge to a stationary strategy.
We are left with a number of unsolved questions, that we plan to explore in the near future. First, we are interested in collecting more detailed data from real-world users, to deepen our understanding of users' interactions and identity revelation processes. For example, in the current study we did not focus on the users' actions, that result in identity disclosure. What are the typical passive social transactions (post an item on your page which may be ÒsilentlyÓ consumed by those who've been given access to it) or active transactions (sharing, commenting on other's content or status updates, give feedback) that lead to information revelation and/or to deception? How do different ÒoutcomesÓ of such transactions affect social capital, and therefore result in truthful and untruthful information sharing? How do secondary (friend-of-friend, triad) relationships influence information sharing? Results obtained from these studies will guide the next step of our research on the model.
For example, using this information, we would like to determine the structural characteristics of the benefit and cost functions in Equation REF . In addition to this, it would be useful to identify whether the dynamics described force the players to converge to an equilibria of some type. We expect that convergence to an equilibrium point should occur, but it is not clear if this is a global property of all well-behaved payoff functions. As noted in [5]} there can be an interaction between the properties of the graph on which the game is played and the number and type of symmetric equilibria. We have not explored this using the model presented in this paper, but we believe this is a necessary step in understanding the behavior of user dynamics in information expression in social networks.

Ackowledgement
Portions of Dr. Griffin's work were supported by the Army Research Office under Grant W911NF-11-1-0487.
